Question

If three distinct number a, b, c are in G.P. and the equations ax² + 2bc + c = 0 and dx² + 2ex + f = 0 have a common root, then which one of the following statements is correct?
(A) d/a, e/b, f/c are in A.P
(B) d, e, f are in A.P
(C) d/a, e/b, f/c are in G.P
(D) d, e, f are in G.P

Answer 1

Answer:

c is correct answer ................

Answer 2

answer : option (A) d/a, e/b, f/c are in ap

given, If three distinct number a, b, c are in G.P. and the equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root.

if a , b, c are in G.P then, b² = ac ......(1)

now, discriminant of ax² + 2bx + c , D = 4b² - 4ac

= 4b² - 4b² = 0 [ from eq (1). ]

so, ax² + 2bx + c has equal roots.

and one of the roots is ; x = -2b/2a = -b/a ,

a/c to question, equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root.

so, -b/a is a root of dx² + 2ex + f = 0

so, d(-b/a)² + 2e(-b/a) + f = 0

⇒db² -2eba + fa² = 0

⇒db² + fa² = 2eba

⇒db² , eba , fa² are In ap

⇒db²/b²a , eba/b²a , fa²/b²a are in ap

⇒d/a, e/b , fa/b² are in ap

putting, b² = ac in fa/b²

i.e., fa/ac = f/c

⇒d/a, e/b , f/c are in ap .